In the given figure, ∠ABC = 90° and BC is diameter of the given circle. Show that : (i) AC × AD = AB 2 (ii) AC × CD = BC 2

(i) As, BC is the diameter. We know that, Angle in a semicircle is a right angle. ∴ ∠BDC = 90° From figure, ⇒ ∠BDC + ∠BDA = 180° [Linear Pair] ⇒ 90° + ∠BDA = 180° ⇒ ∠BDA = 180° - 90° ⇒ ∠BDA = 90°. As, AB is the tangent and BC is diameter and tangent at any point and line from that point to center are perpendicular to each other. ⇒ ∠ABC = 90° In △ABC and △ABD, ∠ABC = ∠ADB (Both equal to 90°) ∠BAD = ∠BAC (Common) ∴ △ABC ~ △ABD In similar triangles, Ratio of corresponding sides are in equal proportion. Hence, proved that AB 2 = AC × AD. (ii) In △ABC and △BDC, ⇒ ∠ABC = ∠BDC (Both equal to 90°) ⇒ ∠BCA = ∠BCD (Common) ∴ △ABC ~ △BDC In similar triangles, Ratio of corresponding sides are in equal proportion. Hence, proved that BC 2 = AD × CD.

In the given figure, AC = AB and ∠ABC = 72°. OA and OB are two tangents. Determine :
(i) ∠AOB
(ii) angle subtended by the chord AB at the center.

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In the given figure, PQ, PR and ST are tangents to the same circle. If ∠P = 40° and ∠QRT = 75°, find a, b and c.

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In the given figure, AB, BC and CA are tangents to the given circle. If AB = 12 cm, BC = 8 cm and AC = 10 cm, find the lengths of AD, BE and CF.

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AB and CD are two chords of a circle intersecting at point P inside the circle. If
(i) AB = 24 cm, AP = 4 cm and PD = 8 cm, determine CP.
(ii) AP = 3 cm, PB = 2.5 cm and CD = 6.5 cm, determine CP.

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Mathematics

In the given figure, ∠ABC = 90° and BC is diameter of the given circle. Show that :
(i) AC × AD = AB²
(ii) AC × CD = BC²

Circles

Related Questions

In the given figure, AC = AB and ∠ABC = 72°. OA and OB are two tangents. Determine :
(i) ∠AOB
(ii) angle subtended by the chord AB at the center.

In the given figure, PQ, PR and ST are tangents to the same circle. If ∠P = 40° and ∠QRT = 75°, find a, b and c.

In the given figure, AB, BC and CA are tangents to the given circle. If AB = 12 cm, BC = 8 cm and AC = 10 cm, find the lengths of AD, BE and CF.

AB and CD are two chords of a circle intersecting at point P inside the circle. If
(i) AB = 24 cm, AP = 4 cm and PD = 8 cm, determine CP.
(ii) AP = 3 cm, PB = 2.5 cm and CD = 6.5 cm, determine CP.

Mathematics

In the given figure, ∠ABC = 90° and BC is diameter of the given circle. Show that :(i) AC × AD = AB2(ii) AC × CD = BC2

Circles

Related Questions

In the given figure, AC = AB and ∠ABC = 72°. OA and OB are two tangents. Determine :(i) ∠AOB(ii) angle subtended by the chord AB at the center.

In the given figure, PQ, PR and ST are tangents to the same circle. If ∠P = 40° and ∠QRT = 75°, find a, b and c.

In the given figure, AB, BC and CA are tangents to the given circle. If AB = 12 cm, BC = 8 cm and AC = 10 cm, find the lengths of AD, BE and CF.

AB and CD are two chords of a circle intersecting at point P inside the circle. If(i) AB = 24 cm, AP = 4 cm and PD = 8 cm, determine CP.(ii) AP = 3 cm, PB = 2.5 cm and CD = 6.5 cm, determine CP.

In the given figure, ∠ABC = 90° and BC is diameter of the given circle. Show that :
(i) AC × AD = AB²
(ii) AC × CD = BC²

In the given figure, AC = AB and ∠ABC = 72°. OA and OB are two tangents. Determine :
(i) ∠AOB
(ii) angle subtended by the chord AB at the center.

AB and CD are two chords of a circle intersecting at point P inside the circle. If
(i) AB = 24 cm, AP = 4 cm and PD = 8 cm, determine CP.
(ii) AP = 3 cm, PB = 2.5 cm and CD = 6.5 cm, determine CP.